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\begin{document}

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\section*{Postnikov system}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{factorization_systems}{}\paragraph*{{Factorization systems}}\label{factorization_systems}

[[!include factorization systems - contents]]

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory}

[[!include (infinity,1)-topos - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{DefinitionForTopologicalSpaces}{For topological spaces}\dotfill \pageref*{DefinitionForTopologicalSpaces} \linebreak
\noindent\hyperlink{DefinitionForSimplicialSets}{For simplicial sets}\dotfill \pageref*{DefinitionForSimplicialSets} \linebreak
\noindent\hyperlink{DefinitionForSimplicialSetsAbsoluteVersion}{Absolute version}\dotfill \pageref*{DefinitionForSimplicialSetsAbsoluteVersion} \linebreak
\noindent\hyperlink{ForSimplicialSetsRelativeVersion}{Relative version}\dotfill \pageref*{ForSimplicialSetsRelativeVersion} \linebreak
\noindent\hyperlink{DefinitionForSimplicialPresheaves}{For simplicial presheaves}\dotfill \pageref*{DefinitionForSimplicialPresheaves} \linebreak
\noindent\hyperlink{constructions}{Constructions}\dotfill \pageref*{constructions} \linebreak
\noindent\hyperlink{ConStructionForSimplicialSets}{For simplicial sets}\dotfill \pageref*{ConStructionForSimplicialSets} \linebreak
\noindent\hyperlink{CoskeletonTower}{Coskeleton tower}\dotfill \pageref*{CoskeletonTower} \linebreak
\noindent\hyperlink{IdentificationRelativeSkeleta}{Identification relative to skeleta}\dotfill \pageref*{IdentificationRelativeSkeleta} \linebreak
\noindent\hyperlink{absolute_postnikov_tower}{Absolute Postnikov tower}\dotfill \pageref*{absolute_postnikov_tower} \linebreak
\noindent\hyperlink{RelativePostnikovTowerConstruction}{Relative Postnikov tower}\dotfill \pageref*{RelativePostnikovTowerConstruction} \linebreak
\noindent\hyperlink{HomotopyClassesRelativeSkeleta}{Homotopy classes relative to skeleta}\dotfill \pageref*{HomotopyClassesRelativeSkeleta} \linebreak
\noindent\hyperlink{ForStrictOmegaGroupoids}{For strict $\omega$-groupoids}\dotfill \pageref*{ForStrictOmegaGroupoids} \linebreak
\noindent\hyperlink{ExamplesForChainComplexes}{For chain complexes}\dotfill \pageref*{ExamplesForChainComplexes} \linebreak
\noindent\hyperlink{in_simplicial_presheaves}{In simplicial (pre-)sheaves}\dotfill \pageref*{in_simplicial_presheaves} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{for_simplicial_sets_3}{For simplicial sets}\dotfill \pageref*{for_simplicial_sets_3} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{postnikov_tower_in_an_category}{Postnikov tower in an $(\infty,1)$-category}\dotfill \pageref*{postnikov_tower_in_an_category} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{Postnikov system} is the [[infinitary factorization system]] of [[(n-epi, n-mono) factorization system|(n-epi, n-mono)-factorizations]] through [[n-images]] in an [[(∞,1)-topos]].

The basic (and historically first) example is the factorization in [[∞Grpd]]/[[Top]] of morphisms to the point: here the Postnikov system assigns to a [[homotopy type]] $X$ a [[tower]]

\begin{displaymath}
X \to \cdots \to X_3 \to X_2 \to X_1 \to X_0  \simeq *
\end{displaymath}
where each $X_k$ is a [[homotopy n-type|homotopy (k-2)-type]]/[[n-truncated object in an (infinity,1)-category|(n-2)-truncated object]] and such that each morphism $X_{k+1} \to X_k$ induces an [[isomorphism]] on [[homotopy groups]] in degree $\leq k-2$. This may be thought of as decomposing $X$ into its ``layers'' as seen by the degree of [[homotopy groups]]. The [[characteristic classes]] which give the [[∞-group extension]] in each layer are called the \emph{[[Postnikov invariants]]} or \emph{[[k-invariants]]} of $X$.

More generally, in any [[(∞,1)-topos]] and for any [[morphism]] $f \colon X \to Y$ the corresponding Postnikov system is a [[tower]]

\begin{displaymath}
f \colon X \simeq im_\infty(f) \to \cdots \to im_2(f) \to im_1(f) \to im_0(f) \simeq Y
\end{displaymath}
of [[n-images]] of $f$ which interpolates between $X$ and $Y$ along $f$: the object $im_n(f)$ is a [[homotopy type]] that looks like $X$ in low degrees (below $n-1$), looks like $Y$ in high degrees (above degree $n$) and looks like the ordinary (1-[[topos theory|topos theoretic]]) [[image]] of $f$ \emph{on homotopy groups} in degree $n-1$ itself.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

In full generality, the Postnikov system is the [[infinitary factorization system]] of [[(n-epi, n-mono) factorization system|(n-epi, n-mono)-factorizations]] through [[n-images]] in an [[(∞,1)-topos]].

The following spells this out explicitly for default [[homotopy theory]], hence in [[∞Grpd]] and in terms of the [[presentable (∞,1)-category|presentation]] by the [[model structure on topological spaces]] and the [[model structure on simplicial sets]]:

\begin{itemize}%
\item \emph{\hyperlink{DefinitionForTopologicalSpaces}{For topological spaces}}


\item \emph{\hyperlink{DefinitionForSimplicialSets}{For simplicial sets}}.



\end{itemize}
With a little bit of care this induces corresponding presentations of Postnikov systems in general [[(∞,1)-topos]] by prolonging to suitable [[model structures on simplicial presheaves]]:

\begin{itemize}%
\item \emph{\hyperlink{DefinitionForSimplicialPresheaves}{For simplicial presheaves}}.

\end{itemize}
For more discussion of the general abstract situation see at

\begin{itemize}%
\item \emph{[[Postnikov tower in an (∞,1)-category]]}

\end{itemize}
\hypertarget{DefinitionForTopologicalSpaces}{}\subsubsection*{{For topological spaces}}\label{DefinitionForTopologicalSpaces}

Historically, Postnikov systems were first described on the model of [[homotopy types]] constituted by [[topological spaces]].

A \textbf{Postnikov system} or \textbf{Postnikov [[tower]]} or \textbf{Moore-Postnikov tower/system} of [[topological spaces]], or rather of their [[homotopy types]], is a sequence of [[path connected topological space|path connected]], [[pointed object|pointed]] [[topological spaces]] $X^{(n)}$, $n \geq 1$, such that $\pi_r(X^{(n)})=0$ for $r \gt n$, together with a sequence $\pi_n$ of [[module]]s of the [[fundamental group]] $\pi_1(X^{(1)})$ of $X^{(1)}$ and [[fibration]]s $p_n : X^{(n+1)} \to X^{(n)}$ classified up to [[homotopy type]] by a specified cohomology class $k^{n+1} \in H^{n+1}(X^{(n)}, \pi_n)$.

\hypertarget{DefinitionForSimplicialSets}{}\subsubsection*{{For simplicial sets}}\label{DefinitionForSimplicialSets}

We discuss the realization of Postnikov systems on [[simplicial sets]]/[[Kan complexes]], first in the

\begin{itemize}%
\item \emph{\hyperlink{DefinitionForSimplicialSetsAbsoluteVersion}{Absolute version}}

\end{itemize}
and then more generally for the

\begin{itemize}%
\item \emph{\hyperlink{ForSimplicialSetsRelativeVersion}{Relative version}}

\end{itemize}
\hypertarget{DefinitionForSimplicialSetsAbsoluteVersion}{}\paragraph*{{Absolute version}}\label{DefinitionForSimplicialSetsAbsoluteVersion}

\begin{defn}
\label{SimplicialPostnikovTower}\hypertarget{SimplicialPostnikovTower}{}
Let $X$ be a [[simplicial set]]. A \textbf{Postnikov tower} for $X$ is

\begin{enumerate}%
\item a sequence

\begin{displaymath}
\cdots \to X_2 
    \stackrel{q_1}{\to}
  X_1
    \stackrel{q_0}{\to} 
  X_0
\end{displaymath}
with maps $i_n : X \to X_n$ such that all [[diagram]]s

\begin{displaymath}
\itexarray{
     && X
     \\
     & {}^{\mathllap{i_n}}\swarrow &&   
      \searrow^{\mathrlap{i_{n-1}}}
     \\
     X_n && \stackrel{q_{n-1}}{\to} && X_{n-1}
  }
\end{displaymath}
[[commuting diagram|commute]];


\item such that for all vertices $v \in X_0$ we have for the [[homotopy group]]s

\begin{displaymath}
\pi_{\gt n}( X_n, v) = 0
\end{displaymath}
and

\begin{displaymath}
(i_n)_* : \pi_i (X,v) \stackrel{\simeq}{\to} \pi_i X_n
\end{displaymath}
for $i \leq n$.



\end{enumerate}
\end{defn}
This appears for instance as (\hyperlink{GoerssJardine}{GoerssJardine, def VI 3.1}).

\hypertarget{ForSimplicialSetsRelativeVersion}{}\paragraph*{{Relative version}}\label{ForSimplicialSetsRelativeVersion}

\begin{defn}
\label{SimplicialPostnikovTowerRelative}\hypertarget{SimplicialPostnikovTowerRelative}{}
Let $f : X \to Y$ be a homomorphism of [[simplicial sets]]. A (relative) \textbf{Postnikov tower} for $f$ is a [[tower]]

\begin{displaymath}
\itexarray{
    X
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    im_\infty(f)
    \\
    \downarrow
    \\
    \vdots
    \\
    \downarrow
    \\
    im_2(f)
    \\
    \downarrow
    \\
    im_1(f)
    \\
    \downarrow
    \\
    im_0(f)
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    Y
  }
\end{displaymath}
that factors $f : X \to Y$ such that for all $n \in \mathbb{N}$

\begin{enumerate}%
\item $X \to im_n(f)$

\begin{enumerate}%
\item induces an [[epimorphism]] on [[homotopy groups]] in degree $n-1$;


\item induces an [[isomorphism]] on [[homotopy groups]] in degree $\lt n-1$



\end{enumerate}

\item $im_n(f) \to Y$

\begin{enumerate}%
\item induces a [[monomorphism]] on [[homotopy groups]] in degree $n-1$;


\item induces an [[isomorphism]] on [[homotopy groups]] in degree $\geq n$.



\end{enumerate}


\end{enumerate}
\end{defn}
This appears for instance as (\hyperlink{GoerssJardine}{Goerss-Jardine, def. VI 2.9}).

\begin{remark}
\label{}\hypertarget{}{}
By the [[long exact sequence of homotopy groups]], the relative Postnikov tower is the tower of the [[(n-connected, n-truncated) factorization system]] of $f$ regarded as a morphism in the [[(∞,1)-category]] [[∞Grpd]]: is is the [[n-epimorphism]],[[n-monomorphism]] factorization through the \emph{[[n-image]]} of a morphism.

\end{remark}
\hypertarget{DefinitionForSimplicialPresheaves}{}\subsubsection*{{For simplicial presheaves}}\label{DefinitionForSimplicialPresheaves}

\hypertarget{constructions}{}\subsection*{{Constructions}}\label{constructions}

We discuss explicit constructions/presentations of Postnikov systems.

\begin{itemize}%
\item \emph{\hyperlink{ConStructionForSimplicialSets}{For simplicial sets}}


\item \emph{\hyperlink{ForStrctOmegaGroupoids}{For strict $\omega$-groupoids}}.



\end{itemize}
\hypertarget{ConStructionForSimplicialSets}{}\subsubsection*{{For simplicial sets}}\label{ConStructionForSimplicialSets}

There are three main functorial models for the Postnikov tower of a [[simplicial set]]:

\begin{itemize}%
\item \emph{\hyperlink{CoskeletonTower}{Coskeleton tower}};


\item \emph{\hyperlink{IdentificationRelativeSkeleta}{Identification relative to skeleta}};


\item \emph{\hyperlink{HomotopyClassesRelativeSkeleta}{Homotopy classes relative to skeleta}}



\end{itemize}
\hypertarget{CoskeletonTower}{}\paragraph*{{Coskeleton tower}}\label{CoskeletonTower}

\begin{prop}
\label{}\hypertarget{}{}
If $X$ is regarded as an [[∞-groupoid]] modeled as a [[Kan complex]], then the [[coskeleton]] sequence

\begin{displaymath}
X = \lim_n cosk_n X \to \cdots \to cosk_{n+1} X \to cosk_{n} X \to \cdots \to *
\end{displaymath}
exhibits a Postnikov tower for $X$.

\end{prop}
This is observed for instance in (\hyperlink{ArtinMazur}{ArtinMazur})) or (\hyperlink{DwyerKan}{DwyerKan}). Also see [[coskeleton]] for more details.

\hypertarget{IdentificationRelativeSkeleta}{}\paragraph*{{Identification relative to skeleta}}\label{IdentificationRelativeSkeleta}

The following construction quotients out the relations encoded by the cells that are thrown in in the above construction, such as to make the maps in the Postnikov tower into [[Kan fibrations]].

We first discuss the absolute tower and then the relative version.

\hypertarget{absolute_postnikov_tower}{}\paragraph*{{Absolute Postnikov tower}}\label{absolute_postnikov_tower}

\begin{defn}
\label{}\hypertarget{}{}
Let $X$ be a [[Kan complex]]. Define for each $n \in \mathbb{N}$ an [[equivalence relation]] $\sim_n$ on the simplices of $X$ as follows: two $q$-simplices

\begin{displaymath}
\alpha, \beta : \Delta^q \to X
\end{displaymath}
are equivalent if their restriction to the $n$-[[simplicial skeleton|skeleton]] coincides

\begin{displaymath}
sk_n(\alpha) = sk_n(\beta)
  : 
  sk_n(\Delta^q)
  \hookrightarrow
  \Delta^q
  \to X
  \,.
\end{displaymath}
Write

\begin{displaymath}
X(n) := X/_{\sim_n}
\end{displaymath}
for the [[quotient]] [[simplicial set]].

\end{defn}
There are evident morphisms

\begin{displaymath}
X(n) \to X(n-1).
\end{displaymath}
\begin{prop}
\label{SimplPostnikovByCoskeletonQuotient}\hypertarget{SimplPostnikovByCoskeletonQuotient}{}
This is a Postnikov tower, def. \ref{SimplicialPostnikovTower}, and all morphisms are [[Kan fibrations]].

Moreover the canonical morphism

\begin{displaymath}
X \to \lim_{\leftarrow_n} X(n)
\end{displaymath}
is an [[isomorphism]], exhibiting $X$ as the [[limit]] (``[[inverse limit]]'') over this tower diagram.

\end{prop}
This appears for instance as (\hyperlink{GoerssJardine}{GoerssJardine, theorem Vi 2.5}).

\hypertarget{RelativePostnikovTowerConstruction}{}\paragraph*{{Relative Postnikov tower}}\label{RelativePostnikovTowerConstruction}

We discuss a model for the relative Postnikov tower, def. \ref{SimplicialPostnikovTowerRelative}.

\begin{defn}
\label{}\hypertarget{}{}
For $f : X \to Y$ a [[Kan fibration]] between [[Kan complexes]], define for each $n$ and each $k$ an [[equivalence relation]] $\sim_n$ on $k$-[[simplices]]

\begin{displaymath}
\alpha, \beta \colon \Delta^k \to X
\end{displaymath}
such that $\alpha \sim_n \beta$ if

\begin{enumerate}%
\item the $n$-[[skeleta]] $sk_n \Delta^k \to \Delta^k \stackrel{\alpha, \beta}{\to} X$ are equal;


\item the [[images]] $f(\alpha), f(\beta)\in Y$ are equal.



\end{enumerate}
Let

\begin{displaymath}
im_{n+1}(f) \coloneqq X/\sim_n
\end{displaymath}
be the simplicial set of [[equivalence classes]] under this equivalence relation.

This canonically comes with morphisms $im_{n_1}(f) \to im_{n_2}(f)$ for $\infty \geq n_1 \gt n_2 \geq 0$.

\end{defn}
For instance (\hyperlink{GoerssJardine}{Goerss-Jardine, def. VI 2.9}).

\begin{prop}
\label{}\hypertarget{}{}
This construction gives indeed a relative Postnikov tower for $f$.

\end{prop}
For instance (\hyperlink{GoerssJardine}{Goerss-Jardine, theorem VI 2.11}).

\hypertarget{HomotopyClassesRelativeSkeleta}{}\paragraph*{{Homotopy classes relative to skeleta}}\label{HomotopyClassesRelativeSkeleta}

For $X$ a [[Kan complex]] and $n \in \mathbb{N}$, let $\tau_{\lt n}X$ be the simplicial set defined as the [[quotient]]

\begin{displaymath}
\tau_{\lt m}X : k \mapsto X_k / homotopy-rel-(n-1)-skeleton
  \,,
\end{displaymath}
where two $k$-cells are identified if there is a simplicial [[homotopy]] between them that fixes their $(n-1)$-skeleton.

This is due to [[John Duskin]]. See for instance (\hyperlink{Beke}{Beke, pages 302-305}).

\hypertarget{ForStrictOmegaGroupoids}{}\subsubsection*{{For strict $\omega$-groupoids}}\label{ForStrictOmegaGroupoids}

\begin{prop}
\label{}\hypertarget{}{}
Consider an object in [[sSet]]/[[∞Grpd]] that is in the image of $Str\omega Grpd$, hence given by a morphism $f \colon X \to Y$ of [[strict ∞-groupoids]].

Then for $n \in \mathbb{N}$ the $(n-1)$-Postnikov stage of $f$ is given by the [[strict ∞-groupoid]] $im_n(f)$ with

\begin{displaymath}
\left(im_n\left(f\right)\right)_k
  =
  \left\{
    \itexarray{
      X_k & \forall  k \lt n-1
      \\
      im(X_{n-1}) \subset Y_{n-1} & \forall \; k = n-1
      \\
      \ast & \forall  k \geq n
    }
  \right.
\end{displaymath}
equipped with the evident [[composition]] operations induced from those of $X$ and $Y$, and equipped with the canonical morphisms of strict $\omega$-groupoids

\begin{displaymath}
X \to im_n(f) \to \ast
\end{displaymath}
(the left one being the identity in degree $k \lt n-1$, the quotent projection in degree $n-1$ and $f$ in degree $k \geq n$, and the right one being $f$ in degree $k \lt n-1$, the image inclusion in degree $n-1$ and the identity in degree $k \geq n$).

\end{prop}
This is discussed in (\hyperlink{BFGM}{BFGM}).

\begin{proof}
The [[homotopy groups]] of a strict $\omega$-groupoid in any degree $k$ are simply given by the groups of $k$-[[automorphisms]] of the identity $(k-1)$-morphism on a given baspoint modulo $(k+1)$-morphisms (hence the homology of the corresponding [[crossed complex]] in that degree). Therefore it is clear from the construction of $im_n(f)$ above that $X \to im_n(f)$ is surjective on $\pi_0$ and an isomorphism on $\pi_{k \lt n-1}$, and that $im_n(f)$ is a monomorphism on $\pi_{n-1}$ and an isomorphism on $\pi_{k \geq n}$.

\end{proof}
\hypertarget{ExamplesForChainComplexes}{}\subsubsection*{{For chain complexes}}\label{ExamplesForChainComplexes}

The following gives a model for the $(n-1)$-stage of a relative Postnikov tower for the special case that the the morphism of Kan complexes is the image under the [[Dold-Kan correspondence]] of a [[chain map]] between [[chain complexes]].

Before we give the explicit formule below as prop. \ref{ImageFactorizationForChainComplexes}, the following remark \ref{ChainComplexnPlusOneImageInDegreen} motivates the formula by regarding chain complexes as models for [[strict ∞-groupoids]] (by this prop.orrespondence\#GlobularAndCubical))

\begin{remark}
\label{ChainComplexnPlusOneImageInDegreen}\hypertarget{ChainComplexnPlusOneImageInDegreen}{}
Let $f_\bullet \colon V_\bullet \longrightarrow W_\bullet$ be a [[chain map]] between [[chain complexes]]

For $n \in \mathbb{N}$, consider the abelian group

\begin{displaymath}
(im_{n+1}(f))_n
    \;\coloneqq\;
  coker(\,
     ker(\partial_V) \cap ker(f_n)
     \to
     V_n
  \,)
\end{displaymath}
For the following it is helpful to think of this abelian group in the following equivalent ways.

Define an [[equivalence relation]] on $V_n$ by

\begin{displaymath}
\left(
     v_n \sim v'_n
  \right)
  \;\Leftrightarrow\;
  \left(
    (\partial_V v_n = \partial_V v'_n)
    \;\text{and}\;
    (f_n(v_n) = f_n(v'_n))
  \right)
  \,.
\end{displaymath}
Then

\begin{displaymath}
(im_{n+1}(f))_n \simeq V_n/_\sim
\end{displaymath}
is equivalently the set of [[equivalence classes]] of this equivalence relation, which inherits abelian group structure since the eqivalence relation is linear.

This is because the equivalence relation says equivalently that

\begin{displaymath}
\left(
     v_n \sim v'_n
  \right)
  \;\Leftrightarrow\;
  \left(
    v_n - v'_n \;\in\; ker(\partial_V) \cap ker(f_n)
  \right)
\end{displaymath}
and hence is generated under linearity by

\begin{displaymath}
\left(
     v_n \sim 0
  \right)
  \;\Leftrightarrow\;
  \left(
    v_n  \in ker(\partial_V) \cap ker(f_n)
  \right)
  \,.
\end{displaymath}
Moreover, notice that the [[Dold-Kan correspondence]]

\begin{displaymath}
DK
  \;\colon\;
  Ch_{\bullet \geq 0}
    \longrightarrow
  KanCplx
\end{displaymath}
factors through [[globe|globular]] [[strict omega-groupoids]] (\href{Dold-Kan+correspondence#GlobularAndCubical}{here}). An [[n-morphism]] in the [[strict omega-groupoid]] $DK(V_\bullet)$ is of the form

\begin{displaymath}
(v_{n-1})
    \overset{\phantom{AA}v_n\phantom{AA}}{\longrightarrow}
  (v_{n-1} + \partial v_n)
  \,.
\end{displaymath}
In terms of these morphisms the [[equivalence relation]] above says that two of them are equivalent precisely if

\begin{enumerate}%
\item they are ``[[parallel morphisms]]'' in that they have the same [[source]] and [[target]];


\item they have the same image under $f$ in the [[n-morphisms]] of $DK(W_\bullet)$.



\end{enumerate}
This suggests yet another equivalent way to think of $(im_{n+1}(f))_n$: it is the [[disjoint union]] over the [[target]] $(n-1)$-cells in $V_{n-1}$ of the images under $f$ of the sets of $n$-cells from zero to that target:

\begin{displaymath}
(im_{n+1}(f))_n
   \simeq
   \underset{v_{n-1} \in V_{n-1}}{\sqcup}
    \left\{
      f_n(v_n) \vert v_n \in V_n \,\text{and}\,\partial v_n = v_{n-1}
    \right\}
  \,.
\end{displaymath}
\end{remark}
\begin{prop}
\label{ImageFactorizationForChainComplexes}\hypertarget{ImageFactorizationForChainComplexes}{}
Let $f_\bullet \colon V_\bullet \longrightarrow W_\bullet$ be a [[chain map]] between [[chain complexes]] and let $n \in \mathbb{N}$. Recall the abelian group $\underset{v_{n-1}}{\sqcup}\{f_n(v_n) \vert \partial v_n = v_{n-1}\}$ from remark \ref{ChainComplexnPlusOneImageInDegreen}.

The following [[diagram]] of [[abelian groups]] [[commuting diagram|commutes]]:

\begin{displaymath}
\itexarray{
    \vdots && \vdots && \vdots
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
    \\
    V_{n+3}
      &\overset{f_{n+3}}{\longrightarrow}&
    W_{n+3}
      &\overset{=}{\longrightarrow}&
    W_{n+3}
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
    \\
    V_{n+2}
      &\overset{f_{n+2}}{\longrightarrow}&
    W_{n+2}
      &\overset{=}{\longrightarrow}&
    W_{n+2}
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
     &&
    \downarrow^{\mathrlap{ \partial_W  } }
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
    \\
    V_{n+1}
      &\overset{f_{n+1}}{\longrightarrow}&
    \left\{
      w_{n+1} | \exists v_n :  \partial_W w_{n+1} = f_n(v_n), \partial_V v_n = 0,
    \right\}
      &\overset{}{\longrightarrow}&
    W_{n+1}
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
     &&
    \downarrow^{\partial_W}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
    \\
    V_n
      &\overset{ (f_n, \partial_V) }{\longrightarrow}&
    \underset{v_{n-1}}{\sqcup}
    \left\{
      f_n(v_n) \vert \partial_V v_n = v_{n-1}
    \right\}
     &\overset{  }{\longrightarrow}&
    W_n
    \\
    \downarrow^{\mathrlap{\partial_V}}
    &&
    \downarrow^{\mathrlap{(f_n(v_n),\partial_V v_n) \mapsto \partial_V v_n}}
    &&
    \downarrow^{\mathrlap{\partial_W}}
    \\
    V_{n-1}
      &\overset{=}{\longrightarrow}&
    V_{n-1}
      &\overset{f_{n-1}}{\longrightarrow}&
    W_{n-1}
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
    &&
    \downarrow^{\mathrlap{\partial_{V}}}
    &&
    \downarrow^{\mathrlap{\partial_W}}
    \\
    V_{n-2}
      &\overset{=}{\longrightarrow}&
    V_{n-2}
      &\overset{f_{n-2}}{\longrightarrow}&
    W_{n-2}
    \\
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
    &&
    \downarrow^{\mathrlap{\partial_{V}}}
    &&
    \downarrow^{\mathrlap{\partial_W}}
    \\
    \vdots && \vdots && \vdots
  }
\end{displaymath}
Moreover, the middle vertical sequence is a chain complex $im_{n+1}(f)_\bullet$, and hence the diagram gives a factorization of $f_\bullet$ into two chain maps

\begin{displaymath}
f_\bullet
   \;\colon\;
  V_\bullet \longrightarrow im_{n+1}(f)_\bullet \longrightarrow W_\bullet
  \,.
\end{displaymath}
Finally, this is a model for the [[n-image|(n+1)-image factorization]] of $f$ in that on [[homology groups]] the following holds:

\begin{enumerate}%
\item $H_{\bullet \lt n}(V) \overset{\simeq}{\to} H_{\bullet \lt n}(im_{n+1}(f))$ are [[isomorphisms]];


\item $H_n(V) \to H_n(im_{n+1}(f)) \hookrightarrow H_n(W)$ is the [[image|image factorization]] of $H_n(f)$;


\item $H_{\bullet \gt n}(im_{n+1}(f)) \overset{\simeq}{\to} H_{\bullet \gt n}(W)$ are [[isomorphisms]].



\end{enumerate}
\end{prop}
\begin{proof}
This follows by elementary and straightforward direct inspection.

\end{proof}
\hypertarget{in_simplicial_presheaves}{}\subsubsection*{{In simplicial (pre-)sheaves}}\label{in_simplicial_presheaves}

The following gives a sufficient condition for modeling [[n-image]] factorizations in some [[(∞,1)-toposes]] with particularly convenient presentation.

\begin{prop}
\label{nImageFactroizationModeledOnSimplicialPrsheaves}\hypertarget{nImageFactroizationModeledOnSimplicialPrsheaves}{}
Let $C$ be a site with [[point of a topos|enough points]], so that the weak equivalences in $sPSh(C)_{\mathrm{loc}}$ are detected on [[stalks]] (\href{model+structure+on+simplicial+presheaves#OverSiteWithEnoughPointsWeakEquivalencesDetectedOnStalks}{this prop.}). Then given a morphism of [[Kan complex]]-valued [[simplicial presheaves]]

\begin{displaymath}
f \colon X \longrightarrow Y$ in $sPSh(C)
\end{displaymath}
such that both $X$ and $Y$ are [[homotopy n-types|homotopy k-types]] for some finite $k \in \mathbb{N}$, then its [[n-image]] factorization in the [[(∞,1)-topos]] $L_{lwhe} sPSh(C)_{loc}$ for any $n \in \mathbb{N}$ is presented by any factorization $X \longrightarrow im_{n}(f) \longrightarrow Y$ in $sPSh(C)$ through some Kan-complex valued simplicial presheaf $im_n(f)$ such that for each object $U \in C$ the [[simplicial homotopy groups]] satisfy the following conditions:

\begin{enumerate}%
\item $\pi_{\bullet \lt n}\left(X(U) \to (im_{n}(f))(U)\right)$ are [[isomorphisms]];


\item $\pi_n\left(X(U) \to (im_{n}(f))(U)\to Y(U)\right)$ is the [[(epi,mono) factorization]] of $\pi_n(f(U))$;


\item $\pi_{\bullet \gt n}\left((im_{n}(f))(U) \to Y(U)\right)$ are [[isomorphisms]].



\end{enumerate}
\end{prop}
\begin{proof}
Evalutation on [[stalks]] is a [[filtered colimit]] which preserves the [[finite limits]] and [[finite colimits]] that go into the definition of [[simplicial homotopy groups]]. Therefore the global conditions assumed on the simplicial homotopy groups imply that the same kind of conditions holds for the stalkwise homotopy groups. These are the [[categorical homotopy groups in an infinity-topos|categorical homotopy groups]] in $L_{lwhe} sPSh(C)_{loc}$. By \href{n-truncated+object+of+an+infinity-category#RecognizngnTuncationOnSimplicialHomotopyGroups}{this prop.} and \href{n-connected+object+of+an+infinity-topos#Connectedness}{this def.} we may recognize $n$-truncation of morphisms on categorical homotopy groups (using the assumption that $X$ and $Y$ are $k$-truncated for some $k$). Therefore the claim now follows from the stalkwise [[long exact sequence of homotopy groups]].

\end{proof}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

It is known that Postnikov systems classify all weak, pointed [[connected space|connected]] [[homotopy type]]s. In particular, if $X$ satisfies $\pi_r(X)=0$ for $1 \lt r \lt n$ then the first non trivial Postnikov invariant is an element $k^{n+1}$ of [[group cohomology]] (with twisted coefficients of course). Such elements are also determined by $n$-fold crossed extensions of $\pi_n$ by $\pi_1$, which are exact [[crossed complex]]es of the form

\begin{displaymath}
0 \to \pi_n \to C_n \to C_{n-1} \to \cdots \to C_2 \to C_1
\end{displaymath}
together with an isomorphism $Coker(C_2 \to C_1) \cong \pi_1$. This gives an algebraic model of such an $n$-[[homotopy n-type|type]]. Advantages of algebraic models are that algebraic constructions can be made on them, such as forming [[limits]] or colimits. The various [[higher homotopy van Kampen theorem]]s are useful in the latter case. For example, it may be difficult or well nigh impossible to write down a determination of the Postnikov invariant of a pushout of crossed modules, even if the pushout consists of finite groups.

A Postnikov system is easiest to understand in the 2-stage case, i.e. two non vanishing homotopy groups, and focuses attention on the cohomology of [[Eilenberg-Mac Lane space]]s, which also determine all [[cohomology operation]]s. Basic work on this area was done by Eilenberg and Mac Lane, and by H. Cartan, while the theory of cohomology operations, including [[Steenrod operation]]s, is itself a large area.

The reference below shows the problems in the 3-stage systems.

For homotopy 3-types, the algebraic model of [[crossed square]]s is more explicit than the corresponding Postnikov system, and more calculable. However, not much work has been done on, say, [[cohomology operation]]s using the algebraic model of $n$-fold groupoids, and it is not clear if that would help.

\hypertarget{for_simplicial_sets_3}{}\subsubsection*{{For simplicial sets}}\label{for_simplicial_sets_3}

\begin{prop}
\label{}\hypertarget{}{}
Let $X$ be a [[Kan complex]] and $\{X(n)\}$ the model for its Postnikov tower from prop. \ref{SimplPostnikovByCoskeletonQuotient}. For any vertex $v \in X_0$ write $K(n)$ for the [[pullback]]

\begin{displaymath}
\itexarray{
    K(n) &\to& *
    \\
    \downarrow && \downarrow^{\mathrlap{b}}
    \\
    X(n) &\to& X(n-1)
  }
  \,.
\end{displaymath}
Let $K(\pi_n (X,v), n)$ be the [[Eilenberg-MacLane object]] on the $n$-[[homotopy group]] of $X$. Then there is a [[weak homotopy equivalence]]

\begin{displaymath}
K(n) \stackrel{\simeq}{\to} K(\pi_n(X,v),n)
  \,.
\end{displaymath}
\end{prop}
This appears for instance as \hyperlink{GoerssJardine}{GoerssJardine, corollary VI 3.7}.

\begin{proof}
Since $K(n) \to K(n-1)$ is a [[Kan fibration]] by prop. \ref{SimplPostnikovByCoskeletonQuotient} the pullback $K(n)$ is the [[homotopy fiber]] of $X(n) \to X(n-1)$.

\end{proof}
\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

There are analogues in other setups, e.g.

\begin{itemize}%
\item [[Postnikov system in triangulated category|Postnikov systems in triangulated categories]]


\item [[motivic homotopy theory]] (M. Levine, \href{http://www.math.uiuc.edu/K-theory/0692}{The Postnikov tower in motivic stable homotopy theory}).



\end{itemize}
\hypertarget{postnikov_tower_in_an_category}{}\subsubsection*{{Postnikov tower in an $(\infty,1)$-category}}\label{postnikov_tower_in_an_category}

We may think of [[Top]] as being the archetypical [[(∞,1)-category]].

In every [[(∞,1)-category]] there is a notion of [[n-truncated object in an (∞,1)-category|n-truncated object]] and accordingly a notion of

\begin{itemize}%
\item [[Postnikov tower in an (∞,1)-category]].

\end{itemize}
The traditional case of Postnikov towers in [[Top]] is a special case of this more general concept.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[tower of homotopy fibers]], [[Adams tower]]


\item [[n-image]], [[n-epimorphism]], [[n-monomorphism]]


\item [[Postnikov invariant]]/[[k-invariant]]


\item [[k-ary factorization system]]


\item [[chromatic tower]], [[Taylor tower]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A standard textbook reference is section 8 of

\begin{itemize}%
\item [[Peter May]], \emph{Simplicial methods in algebraic topology} (\href{http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu}{djvu})

\end{itemize}
and section VI of

\begin{itemize}%
\item [[Paul Goerss]], [[Rick Jardine]], \emph{[[Simplicial homotopy theory]]}

\end{itemize}
Orginal references include

\begin{itemize}%
\item [[M. M. Postnikov]], \emph{Determination of the homology groups of a space by means of the homotopy invariants}, Doklady Akad. Nauk SSSR (N.S.) 76: 359--362 (1951)


\item [[George Whitehead]], \emph{Elements of homotopy theory}, chapter 9


\item Donald W. Kahn, \emph{The spectral sequence of a Postnikov system}, Comm. Math. Helv. 40, n.1, 169--198, 1965 \href{http://dx.doi.org/10.1007/BF02564370}{doi}


\item P. I. Booth, \emph{An explicit classification of three-stage Postnikov towers}, Homology, homotopy and applications 8 (2006), No. 2, 133--155


\item [[G. J. Ellis]] and R. Mikhailov, \emph{A colimit of classifying spaces}, \href{http://www.arxiv.org/abs/0804.3581}{arXiv:0804.3581}.



\end{itemize}
Probably the earliest treatment of Postnikov systems for [[simplicial sets]] is in

\begin{itemize}%
\item [[John Moore|J. C. Moore]], \emph{Semisimplicial complexes and Postnikov systems}, Symposium Internacional de Topologia Algebraica, Mexico City, 1958, pp. 232-247,

\end{itemize}
and as a result, in that context, they are sometimes referred to as \textbf{Moore-Postnikov systems}

The coskeleton construction for the Postnikov tower of a Kan complex is already in

\begin{itemize}%
\item [[M. Artin]], [[B. Mazur]], \emph{\'E{}tale homotopy}, (1969)(Lecture Notes in Maths. 100).

\end{itemize}
Another classical article that amplifies the expression of Postnikov towers in terms of [[coskeleta]] is

\begin{itemize}%
\item [[William Dwyer]], [[Dan Kan]], \emph{An obstruction theory for diagrams of simplicial sets} (\href{http://www.nd.edu/~wgd/Dvi/ObstructionTheoryForDiagrams.pdf}{pdf}), from 1984.

\end{itemize}
Analogous remarks are also in

\begin{itemize}%
\item [[John Duskin]] \emph{Simplicial matrices and the nerves of weak $n$-categories I: Nerves of bicategories} , TAC \textbf{9} no. 2, (2002). (\href{http://www.emis.de/journals/TAC/volumes/9/n10/9-10abs.html}{web})

\end{itemize}
reviewed around page 302 in

\begin{itemize}%
\item [[Tibor Beke]], \emph{Higher ech theory}, K-Theory, 32(4):293--322 (2004) (\href{http://www.math.uiuc.edu/K-theory/0646/}{web})

\end{itemize}
Discussion for [[spectra]] includes

\begin{itemize}%
\item [[Stefan Schwede]], chapter II, section 8 of \emph{[[Symmetric spectra]]}, 2012 (\href{http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf}{pdf})

\end{itemize}
Discussion in [[homotopy type theory]] is in

\begin{itemize}%
\item [[Univalent Foundations Project]], section 7.6 of \emph{[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]}

\end{itemize}
Discussion for [[homotopy types]] modeled by [[crossed complexes]]/[[strict ∞-groupoids]] is in

\begin{itemize}%
\item M. Bullejos, E. Faro, and M. A. Garc\'i{}a-Munoz, \emph{Postnikov Invariants of Crossed Complexes}, Journal of Algebra Volume 285, Issue 1, 1 March 2005, Pages 238--291 (\href{http://arxiv.org/abs/math/0409339}{arXiv:math/0409339}).

\end{itemize}
and for $n$-[[hypergroupoids]] in the thesis,

\begin{itemize}%
\item M. A. Garc\'i{}a-Munoz, 2003, \emph{Un aceramiento algebraico a la theor\'i{}a de Torres de Postnikov}, \href{http://0-hera.ugr.es.adrastea.ugr.es/tesisugr/16555065.pdf}{thesis}, Universidad de Granada.

\end{itemize}
This also contains a good discussion of the link with twisted cohomology and homotopy colimits.

A pedagogical introduction to Postnikov systems with an eye towards their generalization from [[homotopy types]] to [[n-categories]] is in

\begin{itemize}%
\item [[John Baez]], [[Mike Shulman]], \emph{[[Lectures on n-Categories and Cohomology]]}

\end{itemize}
[[!redirects Postnikov tower]] [[!redirects Postnikov towers]]

[[!redirects Postnikov decomposition]] [[!redirects Postnikov systems]]

[[!redirects Postnikov approximation]] [[!redirects Postnikov approximations]]

[[!redirects Moore-Postnikov system]] [[!redirects Moore-Postnikov systems]]



\end{document}